Abstract
In this article, we study the logarithm of the central value \(L\left( \frac{1}{2}, \chi _D\right) \) in the symplectic family of Dirichlet L-functions associated with the hyperelliptic curve of genus g over a fixed finite field \({\mathbb {F}}_q\) in the limit as \(g\rightarrow \infty \). Unconditionally, we show that the distribution of \(\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |\) is asymptotically bounded above by the full Gaussian distribution of mean \(\frac{1}{2}\log \deg (D)\) and variance \(\log \deg (D)\), and also \(\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |\) is at least \(94.27 \%\) Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.
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Acknowledgements
This work was carried out during the tenure of a NBHM Fellowship (funded by DAE) for the first author at ISI Kolkata, India. This work was carried out during the tenure of an NSERC PDF (funded by the government of Canada) for the second author at the Centre de Recherches Math’ematiques, Montr’eal, QC, Canada. The authors also thank the anonymous referees for their valuable comments and insightful suggestions that have improved the quality of the manuscript.
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Darbar, P., Lumley, A. Selberg’s central limit theorem for quadratic dirichlet L-functions over function fields. Monatsh Math 201, 1027–1058 (2023). https://doi.org/10.1007/s00605-023-01853-y
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DOI: https://doi.org/10.1007/s00605-023-01853-y